| Abstract: | In this article, an \({L^p}\)-approach to the primitive equations is developed. In particular, it is shown that the three dimensional primitive equations admit a unique, global strong solution for all initial data \({a \in X_p,D(A_p)]_{1/p}}\) provided \({p \in 6/5,\infty)}\). To this end, the hydrostatic Stokes operator \({A_p}\) defined on \({X_p}\), the subspace of \({L^p}\) associated with the hydrostatic Helmholtz projection, is introduced and investigated. Choosing \({p}\) large, one obtains global well-posedness of the primitive equations for strong solutions for initial data \({a}\) having less differentiability properties than \({H^1}\), hereby generalizing in particular a result by Cao and Titi (Ann Math 166:245–267, 2007) to the case of non-smooth initial data. |
